A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every
isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of
$\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto
$\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal}
if it isometrically embeds every finite metric space $\mathrm{F}$ with
$\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With
$\operatorname{dist}(\mathrm{M})$ being the set of distances between points in
$\mathrm{M}$.)
A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if
it is homogeneous, universal, separable and complete. (It is not
difficult to deduce
that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds
every separable metric space $\mathrm{M}$ with
$\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.)
The main results are: (1) A characterization of the sets
$\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$.
(2) If $R$ is the distance set of an Urysohn metric space and
$\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any
cardinality with distances in $R$, then they amalgamate disjointly to
a metric space with distances in $R$. (3) The completion of every
homogeneous, universal, separable metric space $\mathrm{M}$ is
homogeneous.